As a consequence of both the Onsager reciprocity relations [1]} and the property of the spin-orbit effective fields, the mobility tensor takes the following form in the orthonormal basis \(\lbrace \vec{e}_{x_{\uparrow }},\vec{e}_{y_{\uparrow }}, \vec{e}_{x_{\downarrow }},\vec{e}_{y_{\downarrow }}\rbrace \)  (see Fig.1):
\(\hat{\eta }_{\updownarrow }= \eta \begin{pmatrix}1 & \theta _{so} & 0 & 0\\-\theta _{so} & 1 & 0 & 0 \\0 & 0 & 1 & - \theta _{so} \\0 & 0 & \theta _{so} & 1\\\end{pmatrix}\) 
